They are the irreducible characters of the symplectic and orthogonal groups where the \(x\) 's should be viewed as eigenvalues of the corresponding conjugacy classes of matricesIn the case of the orthogonal groups, the situation depends on whether the group has even or odd rank; we use \(\tilde{X}\)  to stand for eigenvalues from both.. For more on these aspects, we refer the reader to Chapters 24 and onwards of [1]}, as well as the review [2]} and references therein.
